Browse issues
no. 1
Curvature of higher direct images
Philipp Naumann1
1 Philipp Naumann, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 171-201.

Given a holomorphic family f:𝒳→S of compact complex manifolds and a relatively ample line bundle L→𝒳, the higher direct images R n-p f * Ω 𝒳/S p (L) carry induced hermitian metrics. We give an explicit formula for the curvature tensor of these direct images. This generalizes a result of Schumacher in [11], where he computed the curvature of R n-p f * Ω 𝒳/S p (K 𝒳/S ⊗m ) for a family of canonically polarized manifolds. For p=n, the formula coincides with a formula of Berndtsson obtained in [3]. Thus, when L is globally ample, we reprove his result on the Nakano positivity of f * (K 𝒳/S ⊗L).

Étant donné une famille holomorphe f:𝒳→S de variétés complexes compactes lisses et un fibré en droites L→𝒳 relativement ample, les faisceaux images directes R n-p f * Ω 𝒳/S p (L) possèdent des métriques hermitiennes induites. Nous donnons une formule explicite pour le tenseur de courbure de ces images directes. Ceci généralise un résultat de Schumacher dans [11], où il a calculé la courbure de R n-p f * Ω 𝒳/S p (K 𝒳/S ⊗m ) pour une famille de variétés canoniquement polarisées. Dans le cas p=n, la formule coïncide avec la formule de Berndtsson obtenue dans [3]. Donc si L est globalement ample, nous prouvons à nouveau son résultat sur la positivité de f * (K 𝒳/S ⊗L) dans le sens de Nakano.

Received:
Accepted:
Published online:
DOI: 10.5802/afst.1670
Classification: 32L10, 32G05, 14DXX
Keywords: Curvature of higher direct image sheaves, Deformations of complex structures, Families, Fibrations

Philipp Naumann 1

1 Philipp Naumann, Mathematisches Institut, Universität Bayreuth, 95440 Bayreuth, Germany
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
@article{AFST_2021_6_30_1_171_0,
     author = {Philipp Naumann},
     title = {Curvature of higher direct images},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {171--201},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 30},
     number = {1},
     year = {2021},
     doi = {10.5802/afst.1670},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1670/}
}
TY  - JOUR
AU  - Philipp Naumann
TI  - Curvature of higher direct images
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2021
SP  - 171
EP  - 201
VL  - 30
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1670/
DO  - 10.5802/afst.1670
LA  - en
ID  - AFST_2021_6_30_1_171_0
ER  - 
%0 Journal Article
%A Philipp Naumann
%T Curvature of higher direct images
%J Annales de la Faculté des sciences de Toulouse : Mathématiques
%D 2021
%P 171-201
%V 30
%N 1
%I Université Paul Sabatier, Toulouse
%U https://afst.centre-mersenne.org/articles/10.5802/afst.1670/
%R 10.5802/afst.1670
%G en
%F AFST_2021_6_30_1_171_0
Philipp Naumann. Curvature of higher direct images. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 30 (2021) no. 1, pp. 171-201. doi : 10.5802/afst.1670. https://afst.centre-mersenne.org/articles/10.5802/afst.1670/

[1] Constantin Bănică; Octavian Stănăşilă Algebraic methods in the global theory of complex spaces, Editura Academiei; John Wiley & Sons, 1976, 296 pages (Translated from the Romanian) | Zbl

[2] Bo Berndtsson Curvature of vector bundles associated to holomorphic fibrations, Ann. Math., Volume 169 (2009) no. 2, pp. 531-560 | DOI | MR | Zbl

[3] Bo Berndtsson Strict and nonstrict positivity of direct image bundles, Math. Z., Volume 269 (2011) no. 3-4, pp. 1201-1218 | DOI | MR

[4] Bo Berndtsson; Mihai Paun; Xu Wang Algebraic fiber spaces and curvature of higher direct images (2017) (https://arxiv.org/abs/1704.02279)

[5] Jean-Pierre Demailly Complex Analytic and Differential Geometry, 2012 (https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf)

[6] Werner Greub; Stephen Halperin; Ray Vanstone Connections, curvature, and cohomology. Vol. I: De Rham cohomology of manifolds and vector bundles, Pure and Applied Mathematics, 47, Academic Press Inc., 1972, xix+443 pages | MR | Zbl

[7] Phillip A. Griffiths Periods of integrals on algebraic manifolds. III. Some global differential-geometric properties of the period mapping, Publ. Math., Inst. Hautes Étud. Sci., Volume 38 (1970), pp. 125-180 | DOI | Numdam | Zbl

[8] Serge Lang Differential and Riemannian manifolds, Graduate Texts in Mathematics, 160, Springer, 1995, xiv+364 pages | MR | Zbl

[9] Christophe Mourougane; Shigeharu Takayama Hodge metrics and the curvature of higher direct images, Ann. Sci. Éc. Norm. Supér., Volume 41 (2008) no. 6, pp. 905-924 | DOI | Numdam | MR | Zbl

[10] Georg Schumacher The curvature of the Petersson–Weil metric on the moduli space of Kähler–Einstein manifolds, Complex analysis and geometry (The University Series in Mathematics), Plenum Press, 1993, pp. 339-354 | DOI | MR | Zbl

[11] Georg Schumacher Positivity of relative canonical bundles and applications, Invent. Math., Volume 190 (2012) no. 1, pp. 1-56 | DOI | MR | Zbl

[12] Georg Schumacher Erratum to: Positivity of relative canonical bundles and applications [MR2969273], Invent. Math., Volume 192 (2013) no. 1, pp. 253-255 | DOI | Zbl

[13] Yum Tong Siu Curvature of the Weil–Petersson metric in the moduli space of compact Kähler–Einstein manifolds of negative first Chern class, Contributions to several complex variables (Aspects of Mathematics), Volume E9, Vieweg & Sohn, 1986, pp. 261-298 | Zbl

[14] Wing-Keung To; Lin Weng L 2 -metrics, projective flatness and families of polarized abelian varieties, Trans. Am. Math. Soc., Volume 356 (2004) no. 7, pp. 2685-2707 | MR | Zbl

Cited by Sources: